A218276 Convolution of level 2 of the divisor function.

0, 0, 1, 3, 7, 16, 22, 45, 49, 100, 95, 178, 161, 304, 250, 465, 372, 676, 525, 952, 720, 1280, 946, 1702, 1217, 2156, 1570, 2764, 1925, 3376, 2360, 4185, 2912, 4944, 3404, 6121, 4047, 6960, 4858, 8344, 5530, 9600, 6391, 11246, 7513, 12496, 8372, 14926, 9486

Offset

1,4

Comments

Belongs to the family of convolution sums: Sum_{m < n*N} sigma(n)*sigma(n - N*m).

Named W2(n) by S. Alaca and K. S. Williams.

The convolution sum: Sum_{m < n} sigma(n)*sigma(n - m) = W1(n) is A000385(n+1).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

S. Alaca and K. S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.

E. Royer, Evaluating convolutions of divisor sums with quasimodular forms, International Journal of Number Theory 3, 2 (2007), Pages 231-261.

Formula

a(n) = Sum_{m < 2*n} sigma(n)*sigma(n - 2*m).

a(n) = sigma_3(n)/12 + sigma_3(n/2)/3 - n*sigma(n)/8 - n*sigma(n/2)/4 + sigma(n)/24 + sigma(n/2)/24.

a(n) = (1/48)*(22*sigma_3(n) - 2*sigma_3(2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)). - Ridouane Oudra, Feb 23 2021

Maple

with(numtheory): seq((1/48)*(22*sigma[3](n) - 2*sigma[3](2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)), n=1..60); # Ridouane Oudra, Feb 23 2021

Mathematica

Table[Sum[DivisorSigma[1, k]*DivisorSigma[1, n - 2*k], {k, 1, Floor[(n - 1)/2]}], {n, 1, 50}] (* G. C. Greubel, Dec 24 2016 *)

Prog

(PARI) lista(nn) = {for (i=1, nn, s = sum(m=1, floor((i-1)/2), sigma(m)*sigma(i-2*m)); print1(s , ", "); ); }
(PARI) lista(nn) = {for (i=1, nn, v = sigma(i, 3)/12 - i*sigma(i)/8 + sigma(i)/24; if (i%2 == 0, v += sigma(i/2, 3)/3 - i*sigma(i/2)/4 + sigma(i/2)/24); print1(v , ", "); ); }

Crossrefs

Cf. A000385, A218277, A218278.

Sequence in context: A162159 A190890 A116040 * A221025 A036666 A218359

Adjacent sequences: A218273 A218274 A218275 * A218277 A218278 A218279

Keyword

nonn

Author

Michel Marcus, Oct 25 2012

Status

approved